On 9/20/06, David M. Cooke <cookedm at physics.mcmaster.ca> wrote:
>> On Wed, Sep 20, 2006 at 03:01:18AM -0500, Robert Kern wrote:
> > Let me offer a third path: the algorithms used for .mean() and .var()
> are
> > substandard. There are much better incremental algorithms that entirely
> avoid
> > the need to accumulate such large (and therefore precision-losing)
> intermediate
> > values. The algorithms look like the following for 1D arrays in Python:
> >
> > def mean(a):
> > m = a[0]
> > for i in range(1, len(a)):
> > m += (a[i] - m) / (i + 1)
> > return m
>> This isn't really going to be any better than using a simple sum.
> It'll also be slower (a division per iteration). You do avoid
> accumulating large sums, but then doing the division a[i]/len(a) and
> adding that will do the same.
>> Now, if you want to avoid losing precision, you want to use a better
> summation technique, like compensated (or Kahan) summation:
>> def mean(a):
> s = e = a.dtype.type(0)
> for i in range(0, len(a)):
> temp = s
> y = a[i] + e
> s = temp + y
> e = (temp - s) + y
> return s / len(a)
A variant of this can also be used to generate the sin/cos for fourier
transforms using repeated complex multiplication. It works amazingly well.
But I suspect accumulating in higher precision would be just as good and
faster if the hardware supports it.
Divide and conquer, like an fft where only the constant coefficient is
computed, is also a good bet but requires lg(n) passes over the data and
extra storage.
Some numerical experiments in Maple using 5-digit precision show that
> your mean is maybe a bit better in some cases, but can also be much
> worse, than sum(a)/len(a), but both are quite poor in comparision to the
> Kahan summation.
>> (We could probably use a fast implementation of Kahan summation in
> addition to a.sum())
>> > def var(a):
> > m = a[0]
> > t = a.dtype.type(0)
> > for i in range(1, len(a)):
> > q = a[i] - m
> > r = q / (i+1)
> > m += r
> > t += i * q * r
> > t /= len(a)
> > return t
> >
> > Alternatively, from Knuth:
> >
> > def var_knuth(a):
> > m = a.dtype.type(0)
> > variance = a.dtype.type(0)
> > for i in range(len(a)):
> > delta = a[i] - m
> > m += delta / (i+1)
> > variance += delta * (a[i] - m)
> > variance /= len(a)
> > return variance
>> These formulas are good when you can only do one pass over the data
> (like in a calculator where you don't store all the data points), but
> are slightly worse than doing two passes. Kahan summation would probably
> also be good here too.
I think we should leave things as they are. Choosing the right precision for
accumulating sums is absolutely fundamental, I always think about it when
programming such loops because it is always a potential gotcha. Making the
defaults higher precision just kicks the can down the road where the unwary
are still likely to trip over it at some point.
Perhaps these special tricks for special cases could be included in scipy
somewhere.
Chuck
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